# Full text of "Treatise On Analysis Vol-Ii"

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```398       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

For each function /e ^rc(Sp(A/)), we have f(Nn) =f(N) \ En. For this is
immediately seen to be true when/is a polynomial in C and I (15.11.1.1); for
/e ^c(Sp(N))? the result then follows by the Stone-Weierstrass theorem
(7.3.1) and (15.11.1.3); and the general case then follows by (15.11.2).

We shall see that the study of N reduces to that of the Nn. Consider for
example the eigenvalues of N: it is clear that x is an eigenvector of N if and
only if each of its projections xn on En is either zero or an eigenvector of Nn,
corresponding to an eigenvalue independent of n. We have then the
following characterization:

(15.11.6) For aeSp(Nn) to be an eigenvalue of Nn, it is necessary and
sufficient that fj,n ({a}) ^ 0. The corresponding eigenvectors are those belonging
to the image ofEn under the projector (P{^(NH)} and this image is a one-dimen-
sional subfpace.

Let a be an eigenvalue of Nn, and let F be the kernel of od En — Nn. Then
by hypothesis F is closed and nonzero. Also it is stable under TV*, because
NH - (N* • x) = N* * (Nn • x) = ocN* • x. Hence F is stable with respect to the
algebra generated by Nn and N*. If we identify Nn with multiplication by the
class of lc in L^(Sp(Nn), /*„), then F is identified with a subspace of the type
Lc(Sp(Nn), <pM • nn), where M is a universally measurable set such that
/zw(M) 7* 0 (15.10.6). We assert that the support of the measure v = (pM - jurt
consists of a single point. If not, there would exist two disjoint closed subsets
B, C of Sp(Nn), such that v(B n M) ^ 0 and v(C n M) ^ 0. The function
C<PB n Ni(C) (resp. £q>c n M(Q) would be equal almost everywhere (with respect
to v) to a<pBnM(Q (resp. a(pCnM(C))- In other words, we should have C = a
almost everywhere in B n M and almost everywhere in C n M, which is
absurd because B n C = 0. Hence we may assume that M = {/?}, and then
it is clear that Lc(Sp(JVn), (pu - un) is of dimension 1, and that the restriction
of Nn to this space is the homothety with ratio fi.

If a e Sp(JV) is an eigenvalue of N, then the set J of integers n such that a
is an eigenvalue of Nn is not empty. If Dn is the one-dimensional subspace of
En generated by the eigenvectors of Nn corresponding to the eigenvalue a, then
the eigenspace E(a; N) of N corresponding to a (11.1) is the Hilbert sum of
the Dn for neJ. If J is finite, the number of elements of J (which is equal to
the dimension of E(a; N)) is called the multiplicity of a. If J is infinite, then a
is said to have infinite multiplicity. It is easily checked that this notion of
multiplicity agrees with the notion of the multiplicity of a representation
introduced in (15.10.9). Also it follows from the remarks above that if a, /? are
distinct eigenvalues of N, then the eigenspaces E(a; N) and E(/?; N) are
orthogonal.), such that T(gm) • a( tends to 0 and T(gm) • a'n tends to
```